Existence of Localized Pulse Solutions to Skew-Gradient Systems

Reaction-diffusion systems have been primary tools for studying pattern formation. A skew-gradient system is well known to encompass a class of activator-inhibitor type reaction-diffusion systems that exhibit localized patterns such as fronts and pulses. In this dissertation, we investigate standing pulse solutions to two extensions of FitzHugh-Nagumo system that possess a skew-gradient structure. Our models exhibit additional nonlinearities that may enable the models to capture more complex behavior of standing pulse solutions. In both extensions, we employ a variational approach that involves a nonlocal term and establish the existence of standing pulse solutions with a sign change. In addition, we explore some qualitative properties of the standing pulse solutions